In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x.
Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions.
Tarski and Smielew showed that Robinson arithmetic can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).
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Adjoining Elements to a Ring Part 2

Mod01 Lec12 Formal Theories
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References
 Bernays, Paul (1937), "A System of Axiomatic Set TheoryPart I", The Journal of Symbolic Logic, Association for Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862
 Kirby, Laurence (2009), "Finitary Set Theory", Notre Dame J. Formal Logic, 50 (3): 227–244, doi:10.1215/002945272009009, MR 2572972
 Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: NorthHolland Publishing Company, MR 0058532
 Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMS Colloquium Publications, v. 41.